Pseudospectrum

In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators and their eigenfunctions.

The ε-pseudospectrum of a matrix A consists of all eigenvalues of matrices which are ε-close to A:

\Lambda _{\epsilon }(A)=\{\lambda \in \mathbb {C} \mid \exists x\in \mathbb {C} ^{n}\setminus \{0\},\exists E\in \mathbb {C} ^{n\times n}\colon (A+E)x=\lambda x,\|E\|\leq \epsilon \}.

Numerical algorithms which calculate the eigenvalues of a matrix give only approximate results due to rounding and other errors. These errors can be described with the matrix E.

References

Pseudospectra Gateway / Embree and Trefethen

Numerical linear algebra
Key concepts	Floating point Numerical stability
Problems	Matrix multiplication (algorithms) Matrix decompositions Linear equations Sparse problems
Hardware	CPU cache TLB Cache-oblivious algorithm SIMD Multiprocessing
Software	BLAS Specialized libraries General purpose software

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